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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 724 MATHIEU FUNCTIONS following: (a) For a fixed real q, characteristic values a, and b, are real and distinct, if q f O ; aoO and br(q) ap- proach r2 as p: approaches zero. (b) A solution of 20.1.1 associated with a, or b, has r zeros in the interval 0 Iz<'IT, (q real). (c) The form of 20.2.21 and 20.2.23 shows that if azr is a root of 20.2.21 and q is different from zero, then azr cannot be a root of 20.2.23; similarly, no root of 20.2.22 can be a root of 20.2.24 if q # O . It may be shown from other considerations that for a given point (a, q) there can be at most one periodic solution of period 'IT or 2?r if qZ0. This no longer holds for solutions of period S'IT, s23; for these all solutions are periodic, if one is. :It 20 \ -24 FIGURE 20.1. Characteristic Values a,, b, r=0,1(1)5 Power Series for Characteristic Values 20.2.2s + . . . ao(*) =-pz+--- 7q4 29qe 68687q8 2 128 2304'18874368 a1(-q)=1-q--+-----+- q 2 n" q 4 1 1 q 6 49qe 8 64 1536 36864 589824 bl (a) 8348 + , . * 55q7 --- 9437184 35389440 q2 5q4 289q" 12 13824 79626240 bz ( q) =4 - - +- - + . . . 21391pS "458647142400 5p2 763q4 1002401q0 12 13824'79626240 a&)=4+--- - 1669068401q8 + . . . 458647142400 q2 @ 13q4 5q6 16 64 20480 16384 aa(-q)=9+- --+ -+- ba(q) + . . . - 1961q" 609p7 23592960' 104857600 + . . * q2 317q4 10049qe 30 864000+2721600000 b,( a) = 1 6 +- - - + . . .I, p2 433q4 5701q" a4(q) = 16+%+8s4000-272 1600000 2 i i q 4 q 6 as(-q)=25+P+--- 48 774144 147456 b5(d 37 qe '891813888' * q2 1 87 q4 5861633q" b6(~)=36+%+43904000-92935987200000~ * * . + . . . q2 1 87 q4 6743617 p" '36+~+43904000+92935987200000 For r>7, and [ql not too large, a, is approxi- mately equal to b,, and the following approxima- tion may be used 20.2.26 =9+- q2 + (513+7) q4 ",} br 2(fl-1) 32(13-1)3(9-4) + . . . (9r4+58fl+29) qs +64(fl- 1)6(r2-4) (9-9) The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.